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PHYSICAL REVIEW B
VOLUME 59, NUMBER 14
1 APRIL 1999-II
Anisotropic thermodynamics of d-wave superconductors in the vortex state
I. Vekhter
Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
P. J. Hirschfeld
Department of Physics, University of Florida, Gainesville, Florida 32611
J. P. Carbotte
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1
E. J. Nicol
Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
~Received 25 September 1998!
We show that the density of states and the thermodynamic properties of a two-dimensional d-wave superconductor in the vortex state with applied magnetic field H in the plane depend on the angle between H and the
order-parameter nodes. Within a semiclassical treatment of the extended quasiparticle states, we obtain fourfold oscillations of the specific heat, measurement of which provides a simple probe of gap symmetry. The
frequency dependence of the density of states and the temperature dependence of thermodynamic properties
obey different power laws for field in the nodal and antinodal direction. The fourfold pattern is changed to
twofold when orthorhombicity is considered. @S0163-1829~99!50514-6#
The experimental data accumulated over the last few
years have established a consensus that the superconducting
state of the hole-doped high-T c cuprates has a predominantly
d-wave order parameter.1 Such an order parameter possesses
lines of nodes, which results in a gapless excitation spectrum
along certain directions in momentum space. An important
consequence is that the low temperature dependence of thermal and transport properties of the superconducting cuprates
is given by power laws, rather than exponential functions
with an activation energy.2
The properties of the vortex state of d-wave superconductors also differ significantly from those of s-wave materials: while for the s-wave case the density of states ~DOS!
and the entropy are determined at low magnetic fields H
!H c2 by the localized states in the vortex cores, in superconductors with lines of nodes they are dominated by the
extended quasiparticle states, which exist in the bulk along
the nodal directions in momentum space.3 On the basis of
this observation Kübert and Hirschfeld4 suggested a method
of calculating thermal and transport coefficients in the vortex
state microscopically by considering only the contribution of
the extended states and accounting for the effect of the magnetic field on these states semiclassically, via a Doppler shift
of the quasiparticle energy due to the circulating supercurrents. For the field applied perpendicular to the superconducting layers, the supercurrents can be approximated by the
circular velocity field around a single vortex, vs 5 b̂ /2mr,
where r is the distance from the center of the vortex, and we
have set \51. Here b̂ is a unit vector along the current and
we use b as the vortex winding angle. This expression is
valid outside the vortex core and up to a cutoff of order
min$R,l%, where 2R52a 21 AF 0 / p H is the intervortex spacing, F 0 is the flux quantum, a is a geometric constant, and l
0163-1829/99/59~14!/9023~4!/$15.00
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is the penetration depth. Under these assumptions the energy
of a quasiparticle with momentum k is shifted by d v k(r)
5vs •k, and the calculated physical quantities depend on position and have to be averaged over a unit cell of the vortex
lattice. The results obtained within this framework4–7 describe recent experiments well,8–13 suggesting that, while the
effects left outside of this approach are important for a quantitative analysis, the method proposed in Ref. 4 is qualitatively correct and can be used to analyze the properties of the
vortex state of unconventional superconductors.
In this paper we generalize the approach of Ref. 4 to
consider the experimental arrangement with the magnetic
field H in the CuO2 plane, and calculate the density of states.
We find that it exhibits fourfold oscillations as a function of
the direction of the applied field, and that its energy dependence is drastically different for the field directed along the
node and antinode.
Since the c-axis coherence length j c is shorter than the
interlayer distance s, the structure of the vortex state for H in
the plane differs from that with Hic, and is commonly modeled by treating the incoherent c-axis transport as Josephson
tunneling between the layers.14–16 The conclusion of Ref. 3
that the extended states dominate the thermodynamic properties of the vortex state with lines of nodes in the gap remains valid for any orientation of the magnetic field. The
superfluid velocity vs away from the core is governed solely
by the 2p winding of the phase of the order parameter
around each vortex, and, at distances large compared to the
size of the core, is virtually identical to that of an Abrikosov
vortex.14,15 Since the core size is larger than j c , the velocity
field can be approximated by the superflow around a single
vortex only for fields H!H 0 5F 0 / g s 2 , where g is the anisotropy ratio; above that field the cores begin to overlap.15
While for extremely anisotropic cuprates, such as the
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©1999 The American Physical Society
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VEKHTER, HIRSCHFELD, CARBOTTE, AND NICOL
Bi2Sr2CaCu2O81d family, the crossover field is of order of a
few Tesla, for less anisotropic materials, such as
YBa2Cu3O72d , H 0 >50 T, and for all experimentally relevant fields the individual vortices can be treated as Abrikosov vortices in the calculation of bulk properties. Furthermore, in the regime H!H 0 the intervortex distance and the
structure of the vortex state are asymptotically close to those
of an Abrikosov vortex lattice,16 suggesting that the approach of Ref. 4 can be directly applied to the geometry with
the field in the plane. Finally, the c-axis transport remains
incoherent at low temperatures,17 and therefore only the energies of the quasiparticles with momenta in the plane are
relevant to the thermodynamic properties and should be
Doppler shifted.
We now follow the approach of Ref. 4 in neglecting the
contribution of the core states and assuming a spatially uniform order parameter D k over a cylindrical Fermi surface. In
most of this work we consider a pure d-wave angular dependence of the gap, D k[D 0 f ( f )5D 0 cos 2f. We consider a
magnetic field H applied in the a-b plane, at an angle a to
the x axis, and account for its effect on the extended quasiparticle states by the Doppler energy shift d v k(r)5vs •k.
The superfluid velocity is approximated by the flow field of
an isolated vortex, which is elliptical due to the anisotropy of
the penetration depth. We can however rescale the c axis to
make both vs and the intervortex spacing 2R isotropic–in the
London theory this rescaling is z 8 5z(l ab /l c ) –and since the
Fermi surface is two-dimensional there is no associated rescaling of momentum. Then, approximating the unit cell of
the vortex lattice by a circle of radius R, we obtain
d v k~ r! 5
EH
sin b sin~ f 2 a ! ,
r
~1!
where r 5r/R and E H is the energy scale associated with the
Doppler shift
a
E H5 v *
2
A
pH
.
F0
FIG. 1. The density of states at the Fermi level, N 0 ( a ), for
E H /D 0 50.1. The solid curve is the full numerical evaluation of Eq.
~5!, and the dashed ~almost indistinguishable! is the nodal approximation Eq. ~7!.
G ~ k, v n ;r! 52
~ i v n 2vs k! t 0 1D kt 1 1 z kt 3
~ v n 1ivs k! 2 1 z 2k1D 2k
~3!
,
where v n is the fermionic Matsubara frequency, z k is the
band energy measured with respect to the Fermi level, and t i
are Pauli matrices. Standard many-body techniques can be
used to compute physical quantities F(r) at a fixed position
r in real space, and the measured quantities are obtained by
averaging over a unit cell of the vortex lattice according to
^ F & H5
1
p
E r rE
1
2p
d
0
0
dbF~ r,b !.
~4!
We first consider the density of states at the Fermi level
which is easily accessible experimentally via specific-heat
measurements and is given by
N 0 ~ a ! [2
~2!
Here a is a geometric constant of order unity, and in the
London theory the rescaled Fermi velocity v *
5 v f Al ab /l c , where v f is the Fermi velocity in the plane. In
a more general approach v * should be treated as a parameter
related both to v f and the anisotropy of the vortex lattice.
The main difference between geometric arrangements
with the field applied in the plane and that applied along the
c axis is clearly seen from Eq. ~1!. For the field applied
perpendicular to the layers the momentum and real-space
degrees of freedom decouple,4 and the average Doppler shift
is the same at all points on the Fermi surface. In contrast, for
the field in the plane the average Doppler shift becomes dependent on the position at the Fermi surface, and is given by
E H sin(f2a). Since quasiparticles contribute to the density
of states when their Doppler-shifted energy exceeds the local
energy gap, an immediate conclusion is that the density of
states depends sensitively on the angle between the applied
field and the direction of nodes of the order parameter.
To analyze this dependence quantitatively we employ the
single-particle Green’s function which is obtained by introducing the Doppler shift into a BCS Green’s function4
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5
1
Im
2p
1
2p2
(k ^ Tr G ~ k, v 50;r! & H
E fE bE r r
2p
2p
d
F
3Re
1
d
0
0
d
0
u sin b sin~ f 2 a ! u
Asin b sin ~ f 2 a ! 2 ~ D 0 /E H ! 2 r 2 f 2 ~ f !
2
~5!
2
G
.~6!
The integrals over r and b can be done analytically, and the
numerical evaluation of Eq. ~5! is trivial. It has been shown4
that the nodal approximation, which takes advantage of the
fact that the density of states is dominated by the contribution of the regions of the Fermi surface near the gap nodes to
replace vs •k by the Doppler shift at the nodes, vs •kn , provides a remarkably good agreement with the numerical results for T,E H !D 0 . Here it yields
N 0~ a ! .
5
EH
u sin~ f n 2 a ! u
D 0 p nodes
(
2&E H
max~ u sin a u , u cos a u ! .
D 0p
~7!
This result was first obtained by Volovik.18 In Fig. 1 we
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ANISOTROPIC THERMODYNAMICS OF d-WAVE . . .
results remain qualitatively unchanged, although the amplitude of the oscillations is reduced to about 8%, in agreement
with an analytic estimate. This reduction results from incomplete suppression of the contribution to the DOS from the
nodal lines aligned with the field since for quasiparticles outside the equatorial plane the Doppler shift does not vanish.
We suggest that in realistic materials the amplitude of the
oscillations is somewhere between the two estimates and
within the experimental resolution of the specific-heat measurements; the published results suggest that this amplitude
is of order 0.3 mJ/mol K2 at H50.005H c2 i . 18 So far, one
reported measurement performed for two orientations of the
applied field in the plane did not find the predicted
oscillations.8 However, an estimate shows that for the
samples used in Ref. 8 the energy scale E H for H;8 T is
close to the impurity bandwidth, g, which may have resulted
in significant smearing of the fourfold pattern. We also note
that in an orthorhombic system the induced s-wave component of the gap would shift the position of the DOS minimum away from the p /4 direction, and in a heavily twinned
crystal, such as used in Ref. 8, this would result in rapid
filling of the minima and significant suppression of the amplitude of oscillations.
While for a clean sample N 0 ( a )} AH independently of
the angular orientation of H, the energy dependence of the
density of states depends crucially on the direction of the
field. For v ,E H !D 0 ,
FIG. 2. The regions contributing to the density of states for the
antinodal ~a!, and nodal ~b! orientation of the magnetic field.
show the results of full numerical evaluation of the residual
density of states from Eq. ~5! along with the results of Eq.
~7!. The density of states exhibits fourfold oscillations as a
function of the angle of the applied field. There is a broad
maximum for the field applied in the antinodal direction, and
a sharp minimum for the field along the node. Both can be
understood if we notice that the Doppler shift is given by
E 1 5E H u sin(p/42 a ) u at two of the near-nodal regions, and
by E 2 5E H u cos(p/42 a ) u at the other two nodes. When a
field is applied in the antinodal direction, E 1 5E 2 and all
four nodes contribute equally to the density of states, as
shown in Fig. 2~a!. On the other hand, when the field is
applied along a nodal direction, quasiparticles at that node,
which travel parallel to the field, do not contribute to the
DOS; the Doppler shift vanishes at these points. Moreover,
since E H !D 0 , the gap grows faster as a function of the
angle f near the node than the Doppler shift over most of the
unit cell of the vortex lattice, and therefore the quasiparticle
contribution to the density of states is suppressed over the
whole near-nodal region, see Fig. 2~b!. For a field not exactly
in the nodal direction there is always a finite region of the
momentum space where the Doppler shift exceeds the local
gap, resulting in a contribution to the DOS and sharp minima
of N 0 ( a ).
Consequently, for a field in the nodal direction two of the
nodes do not contribute to DOS, while the contribution of the
other two is a factor of & larger than it is for the field along
an antinode. The density of states is therefore reduced by
30%, in agreement with the numerical results. To check how
robust the oscillations are we numerically computed N 0 ( a )
in a three-dimensional superconductor and found that the
N ~ v , a ! . @ N 1 ~ v , a ! 1N 2 ~ v , a !# /2,
~8!
where4
N i~ v , a !
5
5
S
D
1
v
11 2 ,
D0
2x
if x5 v /E i >1;
Ei
@~ 112x 2 ! arcsin x13x A12x 2 # ,
p D 0x
if x<1
~9!
for i51,2, and E 1 and E 2 were defined above. For the field
in the antinodal direction
N ~ v ,0! '
S
D
2&E H
1 v2
11
,
pD0
3 E 2H
~10!
while for the field along a node
N ~ v , p /4! '
v
2E H
1
,
p D 0 2D 0
~11!
see Fig. 3. The frequency dependence of the density of states
follows different power laws for the field along a node or an
antinode, and in the former case the linear slope of
N( v , p /4) does not depend on the magnetic field. Note that
the value of N(0,0) and N(0,p /4) differ by a factor of & as
expected and that the slope of the linear term in Eq. ~10! is
just half the value of the zero-field case as only two nodes
contribute.
Since the frequency dependence of the density of states
determines the temperature dependence of thermal and trans-
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VEKHTER, HIRSCHFELD, CARBOTTE, AND NICOL
FIG. 3. The energy-dependent density of states for different directions of the applied field and E H /D 0 50.1.
port coefficients, our results have profound effects on the
properties of clean d-wave superconductors. In particular, in
addition to the fourfold oscillations of the linear-T term in
the specific heat as a function of the direction of H, the
temperature dependence of C/T AH is linear for the field in a
nodal direction, and quadratic for H away from the node.
The nuclear spin-lattice relaxation time T 1 T and superfluid
density will also exhibit fourfold oscillations and a linear or
quadratic T dependence depending on the direction of the
field. However, if the latter quantity is inferred from the penetration depth measurements, nonlocal effects due to a diverging coherence length19 in the nodal directions may be
important.
Any orthorhombic distortion in the system lifts the fourfold degeneracy of the maxima. One possible way to account
for such a distortion is to consider a d1qs superconductor,
J. R. Kirtley et al., Nature ~London! 373, 225 ~1995!.
J. F. Annett, N. Goldenfeld, and S. R. Renn, in Physical Properties of High Temperature Superconductors II, edited by D. M.
Ginzberg ~World Scientific, Singapore, 1990!, and references
therein.
3
G. E. Volovik, Pis’ma Zh. Eksp. Teor. Fiz. 58, 457 ~1993! @JETP
Lett. 58, 469 ~1993!#.
4
C. Kübert and P. J. Hirschfeld, Solid State Commun. 105, 459
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C. Kübert and P. J. Hirschfeld, Phys. Rev. Lett. 80, 4963 ~1998!.
6
I. Vekhter, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B 59, 1417
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Yu. S. Barash, V. P. Mineev, and A. A. Svidzinskii, Pis’ma
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~1997!#.
1
2
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FIG. 4. The effect of orthorhombicity included in the gap (d
10.15s) on N 0 ( a ) for E H /D 0 50.1. The solid curve is the full
numerical evaluation and the dashed line is the result of the first
equality in Eq. ~7!.
where q!1. Then, even though the position of the nodes is
shifted insignificantly, the fourfold pattern is replaced
by two pairs of peaks with the amplitude ratio
u sin@a20.5 arccos(2q)#u/usin@a10.5 arccos(2q)#u, which differs significantly from unity even for relatively small q, as
shown in Fig. 4. The anisotropy in the density of states and
the thermodynamic properties however remains robust and is
a particularly simple experimental probe of the nodal structure of the order parameter.
We are grateful to K. A. Moler and G. E. Volovik for
important communications. P.J.H. thanks A. Freimuth and R.
Gross for pivotal discussions. This research has been supported in part by NSERC of Canada ~E.J.N. and J.P.C.!,
CIAR ~J.P.C.!, and NSF/AvH Foundation ~P.J.H.!. E.J.N. received partial funding from Research Corporation.
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8